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```Appendix I.

759

seen that hoop-stress diagrams like c -and D will be formed, which
can be calculated if we know the pressure between the surfaces.
It must be noted also that c is compressive and D is tensile hoop
stress. When, however, the fluid pressure is admitted to the
cylinder, it will tend to equalise the stresses, for, supposing E
to be the diagram obtained with a cylinder not initially strained,
the diagram E must, in the actual case supposed, be superposed
on c and D, having regard to sign, or be set up on the base line
abed. The dotted areas will be the final result, and the real
point for us to notice is that, instead of stressing our cylinder to
eft it is only stressed to gh, the outer rings taking their share,
and thus the thickness of the cylinder may be much less than if it
had not been initially strained. The areas c and D will be exactly
equal, because there is equilibrium at first.

A still more equable stress may be obtained (and consequently
less thickness required) by adopting a greater number of rings.

Fig. 749 is a diagram of the stresses when three rings are used,
and will be easily understood from the last diagram. As before,
A = B in area.

In applying these principles to the manufacture of large guns,
botl| the initial and maximum firing tensions are kept within
18 tons per square inch. By using the modulus of elasticity it
will be quite easy to find the hoop stress produced at the ring
between the tubes, and conversely the radial pressure there,
caused by extension due to shrinkage.

When guns are constructed by winding wire very tightly round
a thin core, a perfectly .equable stress may be obtained, and con-
sequently these guns are lightest of all. In Fig. 750, dotted lines
shew tension put in the wire as it is being wound on. The curved
line abc shews resulting stresses in the wire after the gun is-
finished, and the thick line the hoop stresses when the gun is
fired.

Cast Iron Cylinders, if cast without any precaution, will
be in a state of compression on the outside, after cooling, and of
tension on the inside. Building then the hoop-stress diagram,
Fig. 751, upon the incline, we get a very mucji worse result than
before; for the initial stresses, caused by the inside codling last

'\$. 69), only assist the destruction of the cylinder when the```